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Triangles Circumscribed about Central Conics and Their Invariants

Mohammad Hassan Murad

Abstract

We study families of triangles that are inscribed in a fixed circle and circumscribed about a central conic, extending the classical Chapple--Euler relation within the framework of Poncelet geometry. We establish several geometric invariants that arise when the circumcenter of the triangle coincides with either the center of the conic or one of its foci. These include invariance properties of orthic triangles, tangential triangles, polar circles, and trigonometric expressions such as $\sin^2 A + \sin^2 B + \sin^2 C$. We further derive explicit analytic constructions of central conics associated with a given triangle under special configurations, including cases where the foci are located at the circumcenter and orthocenter. In addition, we investigate extremal area problem within Poncelet families, and develop both homothetic and non-homothetic constructions of sequences of Poncelet pairs. These results provide a unified geometric framework linking classical triangle geometry, central conics, and Poncelet porism.

Triangles Circumscribed about Central Conics and Their Invariants

Abstract

We study families of triangles that are inscribed in a fixed circle and circumscribed about a central conic, extending the classical Chapple--Euler relation within the framework of Poncelet geometry. We establish several geometric invariants that arise when the circumcenter of the triangle coincides with either the center of the conic or one of its foci. These include invariance properties of orthic triangles, tangential triangles, polar circles, and trigonometric expressions such as . We further derive explicit analytic constructions of central conics associated with a given triangle under special configurations, including cases where the foci are located at the circumcenter and orthocenter. In addition, we investigate extremal area problem within Poncelet families, and develop both homothetic and non-homothetic constructions of sequences of Poncelet pairs. These results provide a unified geometric framework linking classical triangle geometry, central conics, and Poncelet porism.

Paper Structure

This paper contains 15 sections, 61 theorems, 126 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Properties of Triangles Circumscribed about Central Conics
  3. Constructions of Inellipses with Prescribed Center or Foci
  4. Inellipses with the Center at the Circumcenter
  5. Inellipses with Foci at the Circumcenter and Orthocenter
  6. Explicit Formulas for the Steiner Inellipse and Circumellipse
  7. Homothetic Sequences of Triangles and Associated Central Conics
  8. More Invariants
  9. Orthic Triangles
  10. Polar Circles
  11. Tangential Triangles
  12. Non-Homothetic Sequences of Triangles and Associated Central Conics
  13. Construction of Sequences
  14. Dynamics of the Poncelet Iteration
  15. Areas of Triangles in a Poncelet Family

Key Result

Theorem 1.1

A central conic with semi-minor axis $b$ can be inscribed in a triangle if and only if where $R$ is the circumradius of the triangle, $\varepsilon =1$ (respectively, $-1$) if the conic is an ellipse (respectively, a hyperbola), and $d_{+}$ and $d_{-}$ denote the distances from the circumcenter of the triangle to the foci of the conic.

Figures (15)

  • Figure 1: Proposition \ref{['prop:2.3']} and Corollary \ref{['cor:2.1']}.
  • Figure 2: Proposition \ref{['prop:2.4']}.
  • Figure 3: Proposition \ref{['prop:2.5']}.
  • Figure 4: An ellipse inscribed in $\triangle ABC$ and with center at the circumcenter $O$ of $\triangle ABC$. (Proposition \ref{['prop:3.1']}.)
  • Figure 5: Proposition \ref{['prop:3.2']}.
  • ...and 10 more figures

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • proof
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.1
  • ...and 106 more